莫尔斯透镜:新型一维保角透镜

大多数光学仪器,不论是一个简单的镜头或是复杂的相机镜头,都有各种各样的像差。不过在几何光学中,仍存在无像差的光学仪器,可以使二维或三维空间的所有点成锐像(无像差的像)。这种仪器被称为绝对仪器,在隐身斗篷、超分辨和亚波长聚焦方面有着广泛的应用。传统的光学绝对仪器包括麦克斯韦鱼眼透镜、伊顿透镜、吕内堡透镜以及隐身球透镜,它们的折射率都是在二维或三维空间中依赖半径的函数。

事实上,一维空间中的绝对光学仪器有更深、更本质的物理性质。例如,著名的对称自聚焦米凯尔透镜可以由麦克斯韦鱼眼透镜通过e指数的保角变换得到。与柱对称或者球对称的麦克斯韦鱼眼透镜相比,一维的米凯尔透镜在理论分析和实验制造上更加简单和直观。一维的物理图像也可以很容易地通过保角变换推广到二维或三维,以及没有旋转对称的情况。那么,有一个很自然的问题:除了麦克斯韦鱼眼透镜外,常见的伊顿透镜和吕内堡透镜是否也存在着像米凯尔透镜的一维保角透镜呢?

答案是有。近日,厦门大学陈焕阳教授领导的课题组发现了伊顿透镜和吕内堡透镜之间的关系:他们都可以由一个90年前提出的势——莫尔斯势,通过保角变换得到。相关研究成果发表在Chinese Optics Letters 2020年第6期上(Huanyang Chen, Wen Xiao. Morse lens [Invited][J]. Chinese Optics Letters, 2020, 18(6): 062403)。

莫尔斯势以物理学家菲利普-莫尔斯命名,是量子力学中另一个研究双原子分子间势能相互作用的解析模型。与著名的量子谐振子模型相比,莫尔斯势更加真实,因为它可以描述非谐效应、倍频效应以及组合频率效应。莫尔斯势除了在量子力学中有精确解外,在经典力学中也有振荡形式的位移解。这种量子-经典的对应关系还存在于其他一些传统势上,如库仑势和谐振势,这两类势都在量子力学中有广泛应用,类比到经典几何光学的牛顿势和胡克势,正好对应着伊顿透镜和吕内堡透镜。

该课题组研究人员证明:通过保角变换,莫尔斯势能与库仑势或谐振势相对应。基于莫尔斯势,研究人员提出了一个核心的保角透镜,可以在几何光学中产生伊顿透镜、吕内堡透镜,甚至是更一般形式的透镜。

由莫尔斯势衍生的透镜称为“莫尔斯透镜”,经对数保角变换z=ln w (e指数的保角变换的逆变换)后的透镜称为“广义伊顿/吕内堡透镜”,它是柱对称和球对称的。

在(a)a = -1;(b)a = -2;(c)a = -3;(d)a = -4;(e)a = 1;(f)a = 2;(g)a = 3;(h)a = 4的广义伊顿/吕内堡透镜上,从(1,0)以45°发射的光线的折射率分布(等高线图)和轨迹(黑色曲线)。

通过广义伊顿/吕内堡透镜的折射率表达式可以看出,伊顿透镜和吕内堡透镜是当参数a=-1和a=-2时的特例。模拟光线轨迹的结果表明,伊顿透镜和吕内堡透镜的椭圆轨迹能保角变换到一维的非对称自聚焦轨迹,自聚焦的周期分别为2π和π。由于莫尔斯透镜的折射率分布是不对称的,因此轨迹不再是对称自聚焦,而是非对称的,这与一维的米凯尔透镜不同。此外,如果a取1或者2,即取与伊顿透镜和吕内堡透镜相反形式的值,此时透镜称为“反伊顿透镜”和“反吕内堡透镜”,其成像性质又会有其他新奇的性质。研究人员还发现,不管参数a取整数还是分数,透镜的成像特性都是迥异的,但是它们会遵循着一个共同的规律。

另一个值得一提的趣事是广义伊顿/吕内堡透镜与勒让德-琼斯势(6-12势)的比较。研究人员证明,当a=4时,广义伊顿/吕内堡透镜的势形式和折射率分布与勒让德-琼斯势(6-12势)非常接近,但是它们的成像特性却千差万别:保角的广义伊顿/吕内堡透镜势有4个成像点(自聚焦点),而它的近亲勒让德-琼斯势却无法完美成像。

陈焕阳教授认为:“这项工作为解读量子力学中的势提供了一个经典光学的视角,展示了量子力学结合变换光学在设计光学透镜中的强大能力。所提出的莫尔斯透镜,将会引起研究绝对光学仪器、设计保角变换隐身的研究兴趣。此外,莫尔斯势和莫尔斯透镜的本质物理特性可以将伊顿透镜和吕内堡透镜联系起来。由于伊顿透镜和吕内堡透镜是由伊顿和吕内堡分别单独提出的,因此这是一个新的发现。莫尔斯势丰富和加深了其中的物理意义,这一核心保角透镜将大有可为。”

以上研究中,透镜的尺寸是无限大的,即折射率覆盖了整个空间。如果透镜是有限尺寸的,将有可能设计出一个全向凹透镜,就像吕内堡透镜可以作为一个全向凸透镜一样。另外,广义伊顿/吕内堡透镜和莫尔斯透镜中的轨迹也可能具有类似于麦克斯韦鱼眼透镜和米凯尔透镜的性质。量子力学是一座金矿,其中还存有其他的势如汤川秀树势等,谁将是下一个吃螃蟹的人?

The Morse lens: an intrinsic lens form Eaton and Luneburg lens

Most optical instruments, including a simple lens or sophisticated camera lenses, have various types of aberrations. Nevertheless, there do exist optical instruments that are free of aberrations and can provide sharp (stigmatic) images of all points in certain two-dimensional (2D) or three dimensional (3D) of space within geometrical optics. Such devices are called absolute instruments and have wide applications in cloaks, super-resolution and sub-wavelength focusing. Traditional examples of absolute instruments include Maxwell's fish-eye, Eaton, Luneburg and invisible sphere lenses and their refractive index profile are all radius dependence in 2D or 3D space. In addition, there are more intrinsic physical properties in one dimensional (1D) absolute instrument family. For instance, the well-known symmetric self-imaging Mikaelian lens can be mapped from the Maxwell's fish-eye lens by an exponential conformal mapping w=exp(z). Compared with cylindrical or spherical symmetry Maxwell fish-eye lens, the 1D Mikaelain lens is simpler and more intuitive both in theoretical analysis and experimental fabrication. And the physical images can also be easily extended to 2D or 3D by conformal mappings, even non-symmetric cases. Now there comes a question: Besides the Maxwell's fish-eye lens, whether the common Eaton and Luneburg lenses have such powerful 1D conformal lenses like Mikaelain lens?

The answer is yes. The research group led by Prof. Huanyang Chen from Xiamen University has found the intrinsic relationships between the Eaton lens and Luneburg lens that they can be realized from the Morse potentials proposed about 90 years ago by means of conformal mappings. The research results are published in Chinese Optics Letters, Vol. 18, Issue 6, 2020 (Huanyang Chen, Wen Xiao. Morse lens [Invited][J]. Chinese Optics Letters, 2020, 18(6): 062403).

The Morse potentials, named after physicist Philip M. Morse, is another simple analytic model of the potential energy among diatomic molecules in quantum mechanics. Compared with the famous quantum harmonic oscillator model, Morse potential is more realistic because it describes anharmonic effects, frequency doubling, and combination frequencies. In addition to being precisely solvable in quantum mechanics, the Morse potential has oscillating displacement solutions in classical mechanics as well. This kind of quantum-classical analogy relationship also occurs to some traditional potentials, such as Coulomb potential and harmonic potential. These two potentials are widely used in quantum mechanics and their analogical classical potentials are Newton potential and Hooke potential, which exactly corresponds to Eaton lens and Luneburg lens in geometric optics. In the letter, the researchers prove that, from a conformal mapping aspect, the Morse potential is related to Coulomb potential or harmonic potential, therefore they propose a core conformal lens from the Morse potential and generate an Eaton lens, Luneburg lens, and even the generalized form in geometric optics. The lens originated from the Morse potential is called the Morse lens , and after a logarithmic conformal mapping z=ln w(the inverse transformation of exponential mapping), the lens is called generalized Eaton/Luneburg lens with a cylindrical or spherical symmetry. It is easy to see that the Eaton lens and the Luneburg lens are two special cases when parameter a=-1and a=-2 . With the help of light ray simulations, they find that the familiar elliptical trajectories of Eaton lens and Luneburg lens are mapping to the 1D self-focusing asymmetry trajectories with different periods 2π and π. This is different from the Mikaelain lens, there is not symmetric self-focusing anymore but instead of asymmetric, due to the refractive index profile of the Morse lens. In addition, if parameter a is to take 1 or 2, the opposite case of Eaton and Luneburg lens, named anti-Eaton and anti-Luneburg lens, are proposed with other intriguing properties. They also find that no matter parameter are integers or fractions, the absolute instrument imaging functions are different but obey general rules. Another aspect worth to mention is that the comparison with the famous Lennard–Jones (LJ) potential or 6-12 potential. It is demonstrated that when a=4, even if their potential curves and refractive index profiles are very closed to each other, their imaging properties are opposite: the conformal potentials have four images, however its "close relative" LJ potential, cannot be perfect imaging.

The refractive index distribution (contour map) and trajectory (black curve) for a ray emitted from (1, 0) at 45° on a generalized Eaton/Luneburg lens with (a) a = −1; (b) a = −2; (c) a = −3; (d) a = −4; (e) a = 1; (f) a = 2; (g) a = 3; (h) a = 4.

Prof. Huanyang Chen believes that, "This work provides a classical optics insight for quantum potentials, showing the ability of lensing designs by combining quantum mechanics and transformation optics. The proposed Morse lens, will arise a hot research hit in absolute instrument designs and conformal cloaking designs. The Morse lens and the Morse potentials have more intrinsic physics and can combine Eaton and the Luneburg lens together. It is a novel discovery, because these two lenses are first independently proposed by Eaton and Luneburg. Morse potential enriches and deepens their physical significance and the core of conformal lens will have promising applications."

The lenses above studied are infinite and cover entire spaces, if the lens becomes finite, it is possible to design an omnidirectional concave lens, just like the Luneburg lens can be served as an omnidirectional convex lens. And the trajectories in the generalized Eaton /Luneburg lens and the Morse lens may also be likely to have similar properties to Maxwell's fish-eye lens and Miakelain lens. In another hand, quantum mechanics is a gold mine, there still exist other potentials such as Yukawa potential, and who will be the next to eat crabs?